Goldschmidt, the son of H. J. Goldschmidt, a physical chemist, was born at Zurich in Switzerland. He attended Christiania (now Oslo) University where he obtained his PhD in 1911, remaining in Norway as director of the Mineralogical Institute until 1929 when he moved to the University of Göttingen in Germany. Being a Jew he returned to Norway in 1935, following the rise of anti-Semitism and the Nazi party. He was later sent to a concentration camp but was released by the Norwegian authorities on the grounds of ill health and escaped to England (1942). His time in England was spent first at the Macaulay Institute for Soil Research near Aberdeen, and later at the Rothamsted Experimental Station, Harpenden. He returned to Oslo after the war.

Goldschmidt is acknowledged as the founder of modern geochemistry. Following the work of Max von Laue and W. H. and W. L. Bragg, he laid the foundation for his work by working out the crystal structure of over 200 compounds. His interest was directed to more practical work when, as a result of the naval blockade in the war, he was called upon to investigate Norway's mineral resources.

By the mid-1920s the atomic radii of elements in various stages of ionization had been established. Using this information, together with his detailed knowledge of crystal structure, Goldschmidt began predicting in which minerals and rocks various elements could or could not be found. His results were published over the years in his eight-volume Geochemische Verteilungsgesetze der Elemente (1923–38; The Geochemical Laws of the Distribution of the Elements). His book Geochemistry was published posthumously in 1954.